Optimal. Leaf size=19 \[ -\frac {\sin (e+f x) \cos ^3(e+f x)}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {4043} \[ -\frac {\sin (e+f x) \cos ^3(e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4043
Rubi steps
\begin {align*} \int \cos ^4(e+f x) \left (-4+3 \sec ^2(e+f x)\right ) \, dx &=-\frac {\cos ^3(e+f x) \sin (e+f x)}{f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 31, normalized size = 1.63 \[ -\frac {\sin (2 (e+f x))}{4 f}-\frac {\sin (4 (e+f x))}{8 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 19, normalized size = 1.00 \[ -\frac {\cos \left (f x + e\right )^{3} \sin \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.64, size = 25, normalized size = 1.32 \[ -\frac {\tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 1.45, size = 45, normalized size = 2.37 \[ \frac {-\left (\cos ^{3}\left (f x +e \right )+\frac {3 \cos \left (f x +e \right )}{2}\right ) \sin \left (f x +e \right )+\frac {3 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 33, normalized size = 1.74 \[ -\frac {\tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.35, size = 19, normalized size = 1.00 \[ -\frac {{\cos \left (e+f\,x\right )}^3\,\sin \left (e+f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (3 \sec ^{2}{\left (e + f x \right )} - 4\right ) \cos ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________